Questions: A wooden artifact from an ancient tomb contains 45 percent of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made? (The half-life of carbon-14 is 5730 years.) years

A wooden artifact from an ancient tomb contains 45 percent of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made? (The half-life of carbon-14 is 5730 years.) years

Transcript text: A wooden artifact from an ancient tomb contains 45 percent of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made? (The half-life of carbon-14 is 5730 years.) $\square$ years

Solution

Solution Steps

Step 1: Formula Derivation

The decay of carbon-14 follows an exponential model: $N(t) = N_0 \cdot (1/2)^{t/H}$, where $N(t)$ is the amount of carbon-14 at time $t$, $N_0$ is the initial amount, and $H$ is the half-life of carbon-14.

Given the percentage $P$ of carbon-14 in the artifact compared to living trees, we express $N(t)$ as $P \cdot N_0$. Thus, $P = (1/2)^{t/H}$.

Step 2: Logarithmic Transformation

Taking the logarithm of both sides gives us $\log(P) = \log((1/2)^{t/H})$, which simplifies to $\log(P) = rac{t}{H} \cdot \log(1/2)$.

Step 3: Solving for $t$

Solving for $t$, we obtain $t = rac{\log(P)}{\log(1/2)} \cdot H$.